Explicit diagonalization of the kinship matrix of a pedigree is an unsolved problem in general. In

Explicit diagonalization of the kinship matrix Φ of a pedigree is an unsolved problem in general. In this problem we consider the special case of a nuclear family with n siblings. For convenience, number the parents 1 and 2 and the siblings 3,...,n+2. Let ei be the vector with 1 in position i and 0 elsewhere. Show that the kinship matrix Φ for the nuclear family has one eigenvector e1 − e2 with eigenvalue 1 2 ; exactly n − 1 orthogonal eigenvectors 1 m−3

m−1 j=3 ej − em, 4 ≤ m ≤ n + 2, with eigenvalue 1 4 ; and one eigenvector e1 + e2 +

4λ − 2 n (e3 + ··· + en+2)

with eigenvalue λ for each of the two solutions of the quadratic equation

λ2 − (

1 2 +

n + 1 4 )λ +

1 8 = 0.

This accounts for n + 2 orthogonal eigenvectors and therefore diagonalizes Φ.